Digital communications entail transmitting a bit sequence by modulating a carrier signal onto a carrier wave to assume discrete signal values, or constellation points. While increasing the number of available constellation points allows increased data rates over a given bandwidth, the increase necessarily increases error frequency at the decoder because adjacent constellation points are closer in proximity to one another as compared to a constellation with fewer points. Considering that the decoder uses a maximum likelihood or other probability algorithm to determine exactly which constellation point it has received, the increased error rate is inherent. Trellis coded modulation (TCM) is a coding technique wherein modulation and coding are combined in a manner that reduces error rate by restricting transitions between adjacent constellation points. TCM as referred to herein includes any system that combines a multilevel/phase modulation signaling set with a trellis-coding scheme, or any code system that uses memory (e.g., a convolutional code). A multilevel/phase modulation signaling set is represented by a constellation (other than binary) involving multiple amplitudes, multiple phases, or multiple combinations thereof. A planar example is shown at FIG. 1A, a 16-ary QAM signal constellation.
In an uncoded system, the minimum distance between adjacent constellation points is merely the Euclidean distance. A fundamental concept of TCM systems is that transitions between adjacent constellation points are not allowed. TCM systems allow transitions only between non-adjacent points, so that the minimum Euclidean distance between points in an allowed transition, termed the free Euclidean distance, is greater than the Euclidean distance between two nearest adjacent points. TCM systems can thus increase coding gain without increasing bandwidth, power, or error rate.
The prior art diagrams of FIGS. 1A–1D are instructive. A 16-ary QAM signal constellation 12 of FIG. 1A is divided into mutually exclusive subsets by a series of set partitions, preferably until each subset includes only two points. Assuming that adjacent points of FIG. 1A are separated by the distance d, a first set partition in FIG. 1B yields two subsets 14 and 16 wherein adjacent points are separated by a distance √{square root over (2d2)}. Transitions between the first subset 14 and the second subset 16 are not allowed, so the free Euclidean distance is increased with set partitioning as compared to the uncoded constellation of FIG. 1B. A second set partition is shown in FIG. 1C, wherein each of the subsets 14, 16 of FIG. 1B are divided into two mutually exclusive sets wherein the minimum free Euclidean distance between points is increased to 2d. A third set partition illustrated in FIG. 1D further divides the constellation points among eight subsets wherein the minimum free Euclidean distance between points is increased to √{square root over (8d2)}. Assuming d=1, partitioning into subsets with only two members each yields a free Euclidean distance of 2.828. It is this increase in distance between allowable transitions that enables TCM to increase coding gain (or reduce error rates) without increasing channel bandwidth or power.
Additionally, it is usually assumed that the receiver has perfect knowledge of the channel state for code and constellation design, especially for wireless systems. In a slowly fading channel, where the fading coefficients remain approximately constant for many symbol intervals, the transmitter can send training signals that allow the receiver to accurately estimate the fading coefficients. In this case, one can safely assume perfect channel state information at the receiver, and use codes and constellations that are designed with this assumption. This is termed a coherent communication system. In many practical scenarios, there are some errors in the channel estimates due to the finite length of the training sequence. To maintain a given data rate with errors in the channel estimates, shorter training sequences were required for more rapidly fading channels, resulting in even less reliable channel estimates. Having multiple transmit antennas compounds this problem by requiring longer training sequences for the same estimation performance. Therefore, the usual assumption of known channel parameters at the receiver in designing optimal codes/constellations is not always valid in practice. In the presence of channel estimation errors (partially coherent systems), codes and constellations that are designed using the statistics of the estimation error are more desirable than the ones designed for perfect channel state information at the receiver.